Progress In Electromagnetics Research, PIER 58, 199–222, 2006

ABOUT THE INTERFERENCE INDUCED BYELECTRONS. WHY DOES THE ELECTRON BEHAVEAS A WAVE?

A. Puccini

Via Morghen 155 - 80129 Napoli, Italy

Abstract—One of the most interesting and peculiar phenomena ofQuantum Mechanics is the interference (I) induced by the electrons.Strangely enough, though the electrons are real particles, they oftenbehave just like waves. From the point of view of the classicalmechanics the I induced by the electrons is unexplainable, howeverit is solved mathematically using the formalism of quantum mechanicsand applying Schrodinger’s equation. The quantum solution of theproblem is clear and elegant, especially from a mathematical point ofview, however it still leaves some perplexities as to understand howexactly the phenomenon happens.

We will make a hypothesis trying to understand the undulationphenomenon of the electron: it is really a strange and mysteriousphenomenon. Maybe if we consider that the electron, just as thebaryons and the mesons, might be made of smaller particles (savingthe integrity of the unity of the negative electrical charge and theother Laws of Conservation), we could understand more easily howa single electron can go through two close holes at the same time.Analogously we could better understand another very particularquantum phenomenon carried out mainly by electrons, that is thetunnel effect. In this case, though the particle does not have enoughenergy to go through the potential barrier, though it does not have anymaterial possibility to pass through a layer which does not have anyhole, after several “attempts” the particle will manage to pass throughthe barrier anyway, as it had dug a tunnel, or as it had managed tofind a “breach” in the wall. In this phenomenon too, though we canexplain it from a mathematical point of view, using the equations ofthe quantum mechanics, it is still not clear how actually the electronmanages to have an undulation behaviour.

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Figure 1.

1. INTRODUCTION

One of the most peculiar and interesting phenomena of the QuantumMechanics is represented by the interference (I). We do not mean thephenomenon which takes place when the waves, going through twoslits and without any measurement, make a characteristic I figure ona backstop. What we mean is the I induced by electrons. Strangelyenough, though electrons are real particles, they often behave just aswaves. This does not happen if they have to pass through just one slit,or if we want to be just spectators of the phenomenon. In this casethe I will never take place, the electrons will behave exclusively likeparticles, just as it happens if we shoot normal bullets against a layerwith two slits. Let’s be more clear.

If we use a gun to shoot bullets against a board with two holes,after a certain number of shots, using a detector connected to thebackstop where the bullets arrived, we can calculate the mean of theprobability that the bullets pass through hole 1 (Ph1), or through hole2 (Ph2) (see Fig. 1). It is interesting to notice that “the effect withboth holes open (see Fig. 1d) is the sum of the effects with each holeopen alone (see Fig. 1c). We shall call this result an observation ofno interference” [1]. Indeed if we sum up Ph1 and Ph2 (Fig. 1c),they correspond exactly to the values of Ph1,h2 (Fig. 1d), that is:Ph1 + Ph2 = Ph1,h2.

Following the same experimental criteria, but using a source ofwater waves, as it has been demonstrated several times [2], the resultwill be a typical I figure (see Fig. 2) “if we cover one hole at a time andmeasure the intensity distribution at the absorber we find the rather

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Figure 2.

simple intensity curves shown in part (c) of the figure (2). Ih1 is theintensity of the wave from hole 1 (which we find by measuring whenhole 2 is blocked off) and Ih2 is the intensity of the wave from hole 2(see when hole 1 is blocked). The intensity Ih1,h2 observed when bothholes are open (See Fig. 2d) is certainly not the sum of Ih1 and Ih2. Wesay that there is interference of the two waves. At some places (wherethe curve Ih1,h2 has its maxima) the waves are ‘in phase’ and the wavepeaks add together to give a large amplitude and, therefore, a largeintensity. We say that the two waves are “interfering constructively” atsuch places. There will be such constructive interference wherever thedistance from the detector to one hole is a whole number of wavelengthslarger (or shorter) than the distance from the detector to the other hole.At those places where the two waves arrive at the detector with a phasedifference of π (where they are “out the phase”) the resulting wavemotion at the detector will be the difference of the two amplitudes. Thewaves “interfere destructively”, and we get a low value for the waveintensity. We expect such low values wherever the distance betweenhole 2 and the detector by an odd number of half-wavelengths. The lowvalue of Ih1,h2 in Fig. 2 corresponds to the places where the two wavesinterfere. At the detector the intensity of the wave passing throughhole 1 is proportional to the mean squared height. In the same waythe intensity of the wave coming from hole 2 is proportional to |Ih2 |2.Similarly, for hole 2 the intensity is proportional to |Ih2 |2. When bothholes are open, the intensity is |Ih1 + Ih2|3 [1].

In 1927 Davisson and Germer shot a flux of electrons against alayer with two holes. They got a surprising result. Since electrons areparticles they expected to have a result similar to what happens whenwe shoot bullets, as we can see in Fig. 1. Instead they realised that theimpact position of the electron formed a I (interference) figure typicalof the waves. For the first time it was demonstrated that the flux

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Figure 3.

of electrons behaved, against any prevision, as a wave [2]. ClassicalPhysics was mined: the typical I figure of the waves took place alsowhen electrons were shot one at a time, even one every ten seconds.Later the experiment has been produced several more times; it hasalways confirmed the same results: see Fig. 3 (though it would be thesame if we referred to Fig. 2, since the two graphs are exactly the same).Feynman considers it as “the only mystery of Quantum Mechanics. Aphenomenon which is impossible, absolutely impossible, to explain inany classical way, and which has in it the heart of Quantum Mechanics”[1].

2. DISCUSSION

We have then a particle, the electron, behaving just as a wave. Besides,if it is not observed it can pass simultaneously in both holes! How canjust one particle, of course indivisible, to pass through both holes atthe same time? It makes us think that the electron manages to splitup, but how? It may seem madness, but why cannot we think that theelectron is made of smaller particles? Up until some forty years agothe proton and the neutron were considered elemental particles. LaterGell-Mann’s intuition resulted winning, nowadays we normally teachthat the hadrons are made of quarks (Qs). Namely baryons are madeof three Qs and the mesons are made of a Q and an anti-quark. Whycannot it be the same for the lepton, such as the electron? Of course itis not easy to demonstrate! However the phenomenon of the I inducedby the electron (where a single particle manages to pass through two

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different holes) may represent an “indirect signal” that the electron isnot a “real elemental” particle. How is it made then? It may be madeof 2 (or 3) sub-particles. Awaiting for a possible more suitable word,at the moment we can call these sub-particles “leptoquarks” (LQs).Let’s proceed with order. To introduce this new concept and to makeit possible and/or probable (at least theoretically acceptable) first ofall we need to respect the main “Laws of Conservation” imposed byPhysics. Among these Laws, about a dozen, we consider those morerelated to the electron, considered as an elemental particle, indivisible.Let’s analyze it.1) The Law of Conservation of the Leptopnic Number: the leptonic

number of the electron is +1, it has to remain so. If we considerthe electron as made of two LQs, we have a leptonic number of +2,so we have to assume the presence of an electronic antineutrinoinside the electron, or associated to it. Since the antineutrinohas a leptonic number of −1 we have as a final sum +1, whichcorresponds to the leptonic number of the electron.

2) Law of Conservation of the Electric Charge: as we all know theelectric charge of the electron is −1. Thus, since it has to remainunchanged, we will consider that one LQ has a fractional electriccharge (similarly to the Qs!) of −4/3, whereas the other LQshould, as a consequence, carry an electrical charge of +1/3. Inthis way their sum will give us an electrical charge of −1, that isthe electron’s. Besides, since the electronic antineutrino has a nullcharge, it does not modify the final electrical charge.

3) Law of Conservation of the Angular Momentum: the electron is afermion, indeed it has a half-integer spin of ±1/2. The electronicantineutrino has the same value (spin ±1/2), it should be thesame for each of the two LQs, which should have spins rotatingin contrary directions, so as to balance, without summing up.In this way it is the spin of the antineutrino to rotate in thesame direction of the electron it belongs to. Thus we have a finalhalf-integer spin of ±1/2 — in the same way of the electron —having its same rotation spin (but not a spin of ±11

2). Besideswith the two LQs having anti parallel spins it is respected Pauli’sPrinciple of Exclusion, which forbids two analogous fermions tohave equal quantum state at the same time (though LQs haveelectrical charge differently fractionated and probably differentmasses)

4) Law of Conservation of the Baryonic Number: the electron is alepton, thus it has a null baryonic number, that is zero. It is thesame for the LQs and the electronic antineutrino (since it is alepton too).

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5)–6) Laws of Conservation of the Momentum (p) and of the mass(m): the momentum corresponds to the mass of the consideredparticle multiplied its velocity (p = mv) . In order not to breakthis rule, the probable LQs will have to keep the same momentumof the electron as a sum. Thus, if the two LQs have a massapproximately half of the electron, they should travel in the samedirection and with a speed analogous to the electron; summing upthe momentum of the LQs, it is obtained a value superimposingto the mass and the momentum of the electron they belong to.

7) Law of Conservation of the Energy: it is partially implied in point6), since, as the mass of the electron should distribute more orless equally between the LQs, in the same way, for the Principleof Equivalence Mass-Energy (Einstein), the energy of the electronshould divide in its probable components, however keeping itsenergetic value (0,511 MeV). Besides, the energy associated to theconsidered electronic antineutrino is so small that it does not takea significant quantity of energy from the system.

If we consider the existence of three LQs (just as the number of Qsinside the baryons), things become more complex. In this case wehave a leptonic number of +3, thus in order to respect the law ofconservation of the leptonic number, we need to associate anotherelectronic antineutrino to the electron. Besides, to keep the unity ofnegative charge of the electron, it should not be wrong to think thattwo LQs carry −2/3 electric charge each, and one, on the contrary,carries a charge of +1/3. In this case, in order to respect the Law ofConservation of the Momentum, (as well as Pauli’s Principle) the twoLQs having equal electric charge should show anti parallel spins, thesame should happen with the two antineutrinos. In this way the LQwith an electric charge of +1/3 should have the same spin directionof the electron it belongs to. In this way the law of conservationof the baryonic number would not be broken. As far as the law ofconservation of the momentum is concerned, in the pattern with 3LQs, their speed would not change (respect to the electron), neitherwould change the momentum. The latter would come from the sum ofthe three momentum of the three LQs, each about 1/3 of the electronmomentum in relation to their masses. Moreover it is not taken forgranted that they have an equal mass. We all know that the Qs whichform the most common baryons (2 Qs up and 1 Q down for the proton,or 2 Qs down and 1 Q up for the neutron) have different masses. Thisis true both in the pattern with two LQs and in the one with three LQs.Besides, the way the latter is presented it does not need the “colour”parameter, as it was necessary in the Qs pattern with baryons. Yetthe pattern with 3 LQs appears much more complex than the one with

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2 LQs, since it imposes more sub-particles inside the electron, two ofwhich are electronic antineutrinos: this is a very unusual event, it ismuch less probable. It is implicit that, if what we assumed was true,the positron would be made of two anti-LQs plus an electronic neutrino(or, if ever, by three anti-LQs and two neutrinos).

If what we assumed about the existence of LQs is true, there is onemore aspect to consider. As the Qs are kept together in the hadrons bythe colour force, through the continuous exchange of gluons betweenthe Qs, in the same way we can assume that with electrons electricallycharged the LQs are kept together by a force which we can call “leptonforce” or, in short, leptoforce (LF). According to Quantum Mechanicsthis force would act through a continuous exchange of messengers ofthe force, bosons, which would keep together the LQs in the electron.It is also possible that inside the electron there is a certain freedom ofmovement for the LQs: this “freedom” could explain those “strange”peculiar behaviours of the electron, as when it passes through twoholes at the same time, or through a barrier. We cannot leave outthe so called heavy leptons, such as the muon and the tauon. Theycould be made of heavier LQs, keeping unchanged the total electriccharge. In this way a certain symmetry is kept with heavier Qs. Theheavy leptons, however, belong to the second and third family of theStandard Model, they have a very short life, so we can consider themas extemporaneous particles. The electron, instead, is eternal.

It could be objected: why these probable LQs have never comeout? Why a LQ has never been seen in particle accelerators, or incolliders? The answer could be: for the same reason it will never bepossible to see a free Q, that is a particle carrying a colour charge, butjust through an indistinct jet of hadrons (indeed each Q is eternallyconfined since a few moments after the Big Bang: that is that Q has notcome out from its hadron for 13,7 billion of years). Thus, though theLF cannot be compared to the intensity, to the strength of the strongnuclear force (or colour force), also the LF apparently has not beenprevaricated yet in a particle accelerator, so it has not been possibleyet to point out the probable components of the electron. There isno need to try to describe the behaviour of the probable antineutrinosince, as we all know, it does not leave any sign of its passage (but invery particular conditions).

Following this preamble, since it is certain that a single electroncan go through two holes (close to each other) at the same time, it canbe easier to imagine that it is its probable components that, “loosed”by the LF, manage for a short time to depart from each other (withinthe limits imposed by the LF) so that to go through the holes separatelyand at the same time: one of them through the first hole, the other (or

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the other two) through the second one. It could be a lack of informationof ours, but we do not know such an accurate and detailed experimentfor the passage of the electron through three holes, placed directlyon one first panel (not on a second layer placed behind one with twoholes).

If what we assumed is true, we can examine with Feynman theI induced by the electrons. The latter are shot with an electron gunagainst a thin metal plate with two holes in it. In this experiment too,of course, there is another plate placed beyond the holes, which willserve as a backstop, next to it there is a detector, which might be aGeiger counter, or an electron multiplier connected to a loudspeaker.This allows us to hear all the clicks equal: “there are no half-clicks.The clicks come very erratically, as you have heard a Geiger counteroperating” [1]. It could be seen as a typical quantum phenomenonof randomness, of irregularity, according to the different probabilitieswhere and how the electrons will pass through the holes. “The size(loudness) of each click is always the same. If we lower the temperatureof the wire in the gun, the rate of clicking slows down, but still eachclick sounds the same. We would notice also that if we put two separatedetectors at the backstop, one or the other would click, but never bothat once” [1]. Apparently this is in clear contradiction with what wesaid in the preamble, which is to disprove the fact that the electronmay split up and go through the two holes simultaneously under shapeof its LQs. Indeed, if we shoot the electrons separately, one at a time,we will never hear two simultaneous clicks in the two detectors. Itis not true that the electron splits up then? No, it is, it could bepossible: as soon as it goes beyond the barrier, through the two holesseparately under shape of LQs, the electron “compacts” again justafter, so that “whatever arrives at the backstop it arrives in lumps. Allthe lumps are the same size: only whole lumps arrive, and they arriveone at a time at the backstop. We shall say: electrons always arrivein identical lumps” [1]. It is just as once the particle has fulfilled itsfunction, once it has reached its aim, (i.e., going through some holesin the best way possible, maybe the one with a less waste of energy)it compacts again, it “recombines” (so that to be also less vulnerable,who knows?), arriving whole to the backstop. The curve which weget is just the same of the graph of the waves, represented in Fig. 2.“Yes that is the way electrons go” [1]. Let’s just change the names,let’s substitute the I of the “intensity”, referred to the waves, with theP , indicating the “probability” an electronic lump will arrive at thebackstop (see Fig. 3).

The result obtained with P1,2 (Fig. 3d), that is the graph tracedleaving both holes open, is not the sum of P1 and P2 (Fig. 3c), which

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graph is obtained leaving open one hole at a time. Going along withthe experiment with the waves, also shooting electrons we obtain atypical I pattern (Figs. 2 and 3, coincide perfectly), obtaining on thecontrary a graph different from Fig. 1, when we shoot bullets. Indeed,differently from Figs. 1, “for electrons P1,2 �= P1+P2. How can such anI (interference) come about? Perhaps we should say: well, that means,presumably, that it is not true that the lumps go either through hole1 or hole 2, because if they did, the probabilities should add. Perhapsthey go in a more complicated way. They split in half and .... Butno! They cannot, they always arrive in lumps ...... Well, perhaps someof them go through 1, and then they go around through 2, and thenaround a few more times, or by some other complicated path ....” [1].Well, it should be right so: as the electron passes, it can split in two orthree and go through both slits. It will be argued that it is not possible:“only whole electrons always arrive at the backstop in identical lumps”[1]. Yes, it is true, the electrons arrive at the backstop in lumps, andall the same, but that is because they, probably, combine again in onestructure just as soon as they pass the two slits. It is likely that theLF reunites the two (or three) LQs which probably make the electron:it puts them together, it “reins in”, so they will form a spatial whole,and every single electron will get on the backstop. “But notice! Thereare some points at which very few electrons arrive when both holes areopen, but which receive many electrons if we close one hole, so closingone hole increased the number from the other. Notice, however, that atthe centre of the pattern, P1,2 is more than twice as large as P1+P2. Itis as though closing one hole decreased the number of electrons whichcome through the other hole. It seems hard to explain both effectsby proposing that the electrons travel in complicated paths. It is allquite mysterious. And the more you look at it the more mysterious itseems. Many ideas have been concocted to try to explain the curve forP1,2. None of them can get the right curve for P1,2 in terms of P1 andP2” [1]. Yet, This could be easily explained introducing the conceptthat the electron is made of smaller particles, which can probably splitaway from each other (it should not be a real separation) as they gothrough the two holes separately. In this way it is easier to understandFig. 3d. If we close a hole, the LQs which make each electron do nothave any reason, or any opportunity, to split up, thus the electrons gothrough the open hole all together. The result is what we normallyexpect, adding P1 and P2 (Fig. 3c).

Let’s analyze it mathematically. “Yet, surprisingly enough, themathematics for relating P1 and P2 to P1,2 is extremely simple. ForP1,2 is just like the curve I1,2 of Fig. 2d, and that was simple. Whatis going on at the backstop can be described by two complex numbers

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that we can call φ1 and φ2 (they are functions of x, of course)” [1].The x represents the distance from the centre of the backstop wherethe electron arrives. On their turn φ1 and φ2 are the “probabilityamplitude” that hole 1 or 2 is passed respectively. Whereas theprobability that the event takes place is indicated with the moduleat the power of 2: |φ|2. “The absolute square of φ1 gives the effectswith only hole 1 open. That is, P1 = |φ1|2. The effects with onlyhole 2 open is given by φ2 in the same way. That is, P2 = |φ|2. Themathematics is the same as that we had for the water waves! (it ishard to see how one could get such a simple result from a complicatedgame of electrons going back and forth through the plate on samestrange trajectory). We conclude the following: the electrons arrivein lumps, like particles, and the probability of arrival of these lumpsis distributed like the distribution of intensity of a wave. It is in thissense that an electron behaves sometimes like a particle and sometimeslike a wave” [1]. In conclusion, we learn from mathematics that thevalues of P1,2 are certainly double respect to the result we get from thesum of P1 +P2. Let’s have a look. Let’s represent mathematically thegraph in Fig. 3c:

P1 + P2 = P that is |φ1|2 + |φ2|2 = P

Now, if we simplify further and give the value of 2 to φ, we get:|2|2 + |2|2 = 8. That is P in Fig. 3c gives the result of 8. Whereas

the graph in Fig. 3d, is mathematically expressed as follows:

P1,2 = |φ1 + φ2|2

Since we have given the value of 2 to φ, we will have |2 + 2|2 = 16.The result we get is just double. It is really surprising, but it is whatactually happens. If we try to understand it, this result makes usthink that with both holes open several electrons (or just a part ofthem, who knows?) may split up, divide, in this way going throughboth holes. That is, a single electron, acting through its probablecomponents, manages to pass at the same time through both holes(then it compacts again just before arriving on the backstop). Thisresult, exactly double than what we expect, if our interpretation iscorrect, may make us think that the electron is made of two sub-particles, rather then three (though we cannot be certain about it: wedo not have a mathematical check with three holes).

Let us see now if an electron, splitting up, can actually passthrough both holes at the same time. We use the same experiment,but just add a light source and place it just beyond the holes, at thesame distance from them. Here it comes the so called “measurementparadox”. What is it? If we light the two holes to watch which hole

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the electron comes through, we will always see it pass for one holeat a time, in fact the pattern we get from the backstop is completelyanalogous to the one we have when we shoot bullets. The figure weget in fact is identical to Fig. 1. In short, if electrons are observed,they do not give the I any more. “Here is what we see: every timethat we hear a click from our electron detector (at the backstop), wealso see a flash of light either near hole 1 or near hole 2, but neverboth at once! And we observe the same result no matter where weput the detector. From this observation we conclude that when welook at the electrons we find that the electrons go either through onehole or the other. That is, although we succeeded in watching whichhole our electron comes through, we no longer get the old interferencecurve P1,2, but a new one, P ′

1,2, showing no interference! If we turn outthe light P1,2 is restored. We must conclude that when we look at theelectrons the distribution of them on the screen is different than whenwe do not look. Perhaps it is turning on our light source that disturbsthings? It must be that the electrons are very delicate, and the light,when it scatters off the electrons, gives them a jolt that changes theirmotion. We know that the electric field of the light acting on a chargewill exert a force on it. So perhaps we should expect the motion to bechanged. Anyway, the light exerts a big influence on the electrons. Bytrying to watch the electrons we have changed their motion. That is,the jolt given to the electron, when the photon is scattered by it is suchas to change the electron’s motion enough so that if it might have goneto where P1,2 was at a maximum it will instead land where P1,2 was aminimum; that is why we no longer see the wavy interference effects.When we work up our data (computing the probabilities) we find theseresults: those seen by hole 1 have a distribution like P ′

1; those seen byhole 2 have a distribution like P ′

2 (so that those seen by either hole 1or hole 2 have a distribution like P ′

1,2); and those not seen at all have awavy distribution just like P1,2 of Fig. 3. If the electrons are not seen,we have interference!” [1]. But if the electrons are observed, they donot give the I any longer. “One might still like to ask: how does itwork? What is the machinery behind the law? No one has foundany machinery behind the law. No one can explain any more than wehave just explained. No one will give you any deeper representation ofthe situation. We have no idea about a more basic mechanism fromwhich these results can be deduced” [1]. What happened? Is thistrue only for the electrons or for all particles? Is it a general rule?Apparently it is: it is a consequence of the “paradox measurement”.Thus, If we want to visualize, “to illuminate” the behaviour of aparticle, especially if it is not heavy, we modify its system, since whenwe measure it we determine the “collapse of the wave function” of

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the visualized particle. “A wave function not collapsed involves theidea, not very common, that a particle may be in more places (closeto each other) at the same time, but it is not possible to becomeaware of, because the measurement locates a particle in a position orin another. The concept of the collapse of the wave function goesalong with our experience because it assumes that the measurementforces the wave function to abandon the quantum limbo and chooseone of the several opportunities offered by reality. Thus, if we measurethe position of the electron, for instance, we make its wave functionchange shape suddenly. When we measure the position of the electronits peaks collapse, reducing to zero in all places where the particle arenot, the probabilities increase to 100% in the only position where theparticle is found with the measurement” [2]. That is why the figureof I disappears: the amplitude probabilities, related to the positionwhere the electron could have been found (or its probable components),were compressed as the particle was observed, they reduced in oneposition. This is the “collapse of the wave function” or “reduction ofthe amplitudes”.

From what we have assumed, we can infer that the measurement,the visualization, that is lightening the electrons through the photonsof the visible band, can bring the LQs near to each other, so that thedistance between them is reduced (who knows? It may coincide withthe “reduction of the amplitudes”). It is as though with an externalinput on the electron, the LF “alerted” and “assembled” the LQs,packing them in a narrow space. Why? Who knows? Maybe wecan talk about something like a “surviving instinct” of the particle: asthough, “touched” by a foreign body, as a sufficiently energetic photon,it may feel “threatened” and thus “withdraw into itself”. Who knows?Yet the collapse of the wave function is something real. We quotethe interpretation given by Heisenberg: we do not know where theelectron is before the measurement, this lack of knowledge reflects onthe wave function, which describes it potentially present in differentpoints. When we measure its position, our knowledge of it suddenlychanges: in theory now we know where it is with accuracy. The suddenchange of our knowledge determines as much sudden a change in thewave function which collapses and assumes the configuration of onecrest” [2]. Therefore the sudden collapse of the wave function is not atall a surprising phenomenon, it is just a sharp and sudden change at aknowledge level [2].

On the other hand without measurement we have the I: it is asthough the electron, not disturbed, kept the LQs with “slacken reins”,as though the string tight by the LF between the LQs was looser, (as ithappens with the Qs in baryons, in the same way as the higher or lower

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“tension” used by the colour force through the gluons). Thus, since theLQs are freer to move, those belonging to the same electron manageto go through the plate, passing simultaneously each for one hole, thisexplains the I we get. Somebody may ask: “why should it have such abehaviour? In case LQs do exist, why should they pass each for a hole?What is the goal? In case there is one. 1) The phenomenon of I issomething real. 2) The I is a typical phenomenon of the wave, whereasit is a mysterious peculiarity which occurs with the electron, which isa particle, has a matter, a certain weight, though a light one. 3) Ifwe visualize the electrons when they go through one hole or the other,just because of the “measurement paradox” the electrons pass throughone hole at a time and we will not have the I. 4) if the electron is notdisturbed, if the physical system is not modified, we will have the I.5) a single undisturbed electron will be able to pass through the twoholes simultaneously (contributing to the I). Maybe the most difficultpoint to explain is number 5. Maybe we can hope to understand thisphenomenon introducing the concept of the electron being made of 2–3minor particles. It is certain that the electrons arrive on the backstopas a whole. But as we said, once the electron, the particle, has gonebeyond the obstacle, probably it does not have any reason to be spreadin the space, thus it compacts again and allows us to detect it on thebackstop. Why should it recombine? Who knows? May it dependon some natural rule innate in the microscopic world? Maybe we canconsider it as a spontaneous behaviour rather than as a rule. Is it thespontaneous behaviour to follow the easiest and most linear path, theone with a “minor waste of energy”? May it also depend on quantumfluctuations inside the electron or on the physical composition andequilibrium of the electron, a peculiarity of the LF, which sometimeskeeps the LQs more bound together and sometimes make them spread?It is as though when the electron is about to go through the layer withtwo close slits, the “energetic string” represented by the LF wouldstretch out, as the string of a bowstring. Thus the LQs — placed atthe two ends of the string — manage to pass through the two holesat the same time. After that, when the necessity ends, the stringloosens again, it is not stretched any more, so that the LQs gatherand compact again. Who knows? It is really hard to say. We canjust say that a similar behaviour is likely to happen with Qs, when the“energetic string” which keeps them together, represented by the colourforce, is more stretched or more loose according to the necessities.Besides, the idea that the electron is not as simple as it appears is notsomething new. Protons and neutrons seem elemental particles too.Schrodinger already sensed some complexity about it. “Schrodingeradvanced a hypothesis: maybe the substance the electrons are made

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can be distributed in the space, forming the material support to thewave with which the electron can show itself. [2]. Why cannot we thinkthat “such a substance of which electrons are made” is represented justby the LQs?

Besides, “in the last 80 years it has been proved the usefulnessof the wave functions in the prevision and interpretation of severalexperimental data, nevertheless, there is not yet a way universallyaccepted to define them at a physical level: it is still a controversialpoint that the wave function of the electron is the electron itself, isassociated to the electron and is a mathematical instrument to show themotion of the electron or the representation of what we can know aboutit. However what we know is that thanks to these waves the quantummechanics has introduced, in an unexpected way, the probability inthe physical laws” [2].

Anyway, the main point is to try to explain the I of the electron.“The interference effect occurs also when there is only one electron ineach moment. A single electron can show the I. It can go through bothslits and interfere with itself, if we may say so” [3]. Several experimentshave shown that an elemental particle such as the electron, has alsoan undulation character. “The typical interference figure of the weavesoccurs also if we shoot the electrons separately, one by one, against thescreen” [2]. If what we assumed is true, maybe they are the particlesmaking the electron (which move inside it and are kept together bythe LF) to propagate as waves: so some of them can pass through aslit some through the other. Thus, if two LQs separate to go eachthrough a hole, it can be thought that the “energy tail” which keepsthem together in the electron, represented by branches of the LF,may temporarily break and come together once it has gone throughthe holes. This tail should be “virtual”, that is not material, butenergetic, since it is represented by a continuous flux of bosons whichLQs exchange continuously. It should be just this continuous exchangeof bosons, which we can call “leptonic bosons”, to represent the LF.That is there may be a more or less strict analogy with the gluons ofthe colour force: also in that case when the string tightens too much(as a consequence of an excessive removal of the Qs, within the limitsof their freedom of movement) it breaks and comes together right after.After going through the holes separately, once the link between the LQsis fully restored, the electron compacts again and arrives as a wholeon the backstop, equal to all the other electrons arriving there. Thatis the probability amplitude may tell us where to find the fundamentalcomponents of the particle, they move continuously inside the particlebut we do not know in which part of the “field” of the particle theyare in a specific moment. The wave function can help us foresee them

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with “probability”. Besides, “it is not possible to see the wave functionsdirectly” [2], maybe just because it is not possible to see directly theLQs, nor it is possible to isolate them (as it happens with the Qs,which can be detected only indirectly, statistically, through a jet ofhadrons) [4–6]. We all know that the wave functions can be describedmathematically, applying to Schrodinger’s Equation (1926):

Hψ(x, t) = ihdψ(x, t)dt

,

where H is the Hamiltonian, h is the Planck’s constant reducedor rationalized, corresponding to h/2π, i is the imaginary unity, xand t are respectively the spatial and temporal coordinates, d is thederivation index and ψ is the wave function. The latter comes from DeBroglie’s intuition (1923), according to which, likewise electromagneticradiations, material corpuscles, and particularly electrons, can in away be associated to a wave. De Broglie’s relation is the following:p = h

λ where p is the impulse or the momentum of the electron, his the Planck’s constant and λ is the wave length of the particle.It is interesting to note that the De Broglie equation is exactly thesame when we want to express mathematically the impulse of anelectromagnetic wave (the photon). Besides, it can be useful tospecify that: “it was Max Born who correctly interpreted the ψ ofthe Schrodinger equation in terms of a probability amplitude — thatvery difficult idea that the square of the amplitude is not the chargedensity but is only the probability per unit volume of finding anelectron there, and that when you do find the electron some placethe entire charge is there” [7]. Finding the electron means modify thesystem (as the measurement), thus the result is the collapse of the wavefunction. Hence the probable components of the electron condensateimmediately, that is they gather in one place, concentrating in a verynarrow space the entire electrical charge of the particle, whereas theother points of the field of the electron tend to zero, they disappear.Yes, we specified namely the field of the electron. Indeed, “It is possibleto apply the idea of the field to the matter too. In a certain way thewave function can be interpreted as fields which action is to providethe “probability” that a certain material particle is in a certain regionof the space. We can think of an electron as a particle (the one whichleaves a mark on a phosphor screen or on backstop), but it can bethought or rather it has to be thought as a wave field too, whichgenerates interference phenomena. There is a way to associate to thewave function of the electron its field (it is called electron field) similarlyto the electromagnetic field where the role of the photon, however, isplayed by the electron itself” [2]. If the electron is made of smaller

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particles (the LQs), kept together by the LF, it is possible to think,instead, that it is the electron itself to make in its whole the “electronfield”. Besides, it seems quite incongruous that a particle interactswith an identical particle, using as a boson the same kind of particle. Itcould be much more logical, congruous and natural that the messengerof the electron field is the “leptonic boson” (corresponding to the gluonamong the hadrons).

From what we said, it is also important “to emphasize a veryimportant difference between classical and quantum mechanics. Wehave been thinking about the probability that an electron will arrivein a given circumstance. We have implied that in our experimentalarrangement (or even in the best possible one) it would be impossibleto predict exactly what would happen. We can only predict the odds!This would mean, if it is very true, that physics has given up on theproblem of trying to predict exactly what will happen in a definitecircumstance: Yes! Physics has given up. We do not know how topredict what would happen in a given circumstance, and we believenow that it is impossible that the only thing that can be predicted isthe probability of different events. It must be recognized that this isa retrenchment in our earlier ideal of understanding nature. It maybe a backward step, but no one has seen a way to avoid it. We makenow a few remarks on a suggestion that has sometimes been made totry to avoid the description we have given: perhaps the electron hassome kind of internal works — some inner variables — that we donot yet know about. Perhaps that is why we cannot predict what willhappen. If we could look more closely at the electron, we could be ableto tell where it would end up. So far as we know, that is impossible.We could still be in difficulty. Suppose we were to assume that insidethe electron there is some kind of machinery that determines where itis going to end up” [1]. Thus we know that Feynman too started to“suspect” that the electron was not such a simple particle, elemental,he suspected that “inside the electron there could be some kind ofmachinery, some internal variable”, able to guide the path. Therefore,our hypothesis would not be conceptually too far. “That machinerymust also determine which hole it is going to go through on its way”[1]. We may say that the electron follows the way which seems theeasiest, the most linear and the most natural. If it finds only one holeopen, it will go through this one. If it finds two open holes, it willprobably, most of the times, make the I. How? This is the mainquestion which inspired this article. It is very well proved by nowthat the electron induces the I, just as a wave, yet it is a particle.But this is explainable; it happens with other particles too, De Broglieguessed that as early as 1924. Besides, the energy of the electron

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corresponds to an electromagnetic wave of a certain frequency such asthe γ ray, in which it transforms when it collides with its antiparticle.In short, the fact that the electron behaves like a wave does not causeany astonishment! We know this. Yet one may wonder: how can asingle particle — which is considered “elemental”, that is it cannot bedivided in further smaller particles — pass through two different holes?As it had the gift of ubiquity. It seems more logical to infer thatthe electron splits temporarily in its probable components. The latterwill go through the two holes contemporarily: a LQ through a hole,the other (or the other two) through the second one, then they willcompact again in the electron and get on the backstop as a whole, asone particle. It could be an inborn characteristic of the electron to beable to go through two holes contemporarily, if it is true that when theelectron travels the whole electron system, with its field and contents,moves. We could think that it is the electron field to go through thetwo holes contemporarily: it would precisely pass through them as a“field”. Besides we read: “But we must not forget that what is insidethe electron should not be dependent on what we do, and in particularupon whether we open or close one of the holes. So if an electron,before it starts, has already made up its mind which hole it is going touse, and where it is going to land” [1]. Why should the electron havedecided a priori which hole go through? It should be more likely thatalong the way the electron “feels” which way is easier; maybe it willrespect the law of the minor waste of energy — “the Principle of LeastAction” [8] — following the closest path, just as the photon follows theshortest and straightest path: “that out of all possible paths that itmight take to get from one point to another, light takes the path whichrequires the shortest time” (Fermat’s Principle of Least Time, 1650)[9]. Or, in case the electron finds all “blocked up”, it will probablytry, with “great effort”, that is with remarkable waste of energy, afterseveral attempts, to exploit the “tunnel effect”. Let’s analyze shortlythis phenomenon.

The “tunnel effect” (TE) is one of the oddest properties of thequantum world. It is a typically quantum effect: it cannot becalculated and justified applying the laws of the classical physics, theNewtonian one. The problem that this effect manages to solve isthe calculus of probability that a particle, having a certain energy,manages to get over a barrier with an energetic potential higher thanthe energy of the particle itself. In this case, in fact, the laws ofclassical mechanics immediately give us a null result of probability.On the contrary, the formalism of the quantum mechanics allows usto calculate it confirming the experimental reality. Also in this caseit is useful to apply Schrodinger equation, otherwise the phenomenon

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would be unexplainable: from the calculus it appears the probability,small but well defined, to penetrate the barrier and, if this is notinfinite in height and thickness, to pass through it and get beyondit. It is important to mention that the lower the height and/or thethickness of the barrier, as well as the lighter the particle, the higherthe penetrability. This is why the electron is the most suitable particleto use the TE. It is because sub-atomic particles do not have to beconsidered as rigid objects, but a sort of clouds (the “electronic cloud”,the electron field), thus they are able to go through a thin barrier.

Thus, using the TE a particle, an atomic nucleon, has a probabilitynot null to pass through a potential barrier even though the particlehas a kinetic energy lower than the maximum height of the potentialbarrier. That is, a barrier which is absolutely impenetrable froma classical point of view (a particle which bumps against a barrierwithout having enough energy to pass it will not be able to gothrough it in any way), becomes penetrable for the quantum mechanics.What happens is that the penetration of a potential barrier, whichis unexplainable from a corpuscular point of view, becomes possibleas a undulation phenomenon, that is when the particle leaves itscorpuscular aspect and gets a typically undulation behaviour. Allthings considered, it seems that the particle manages to go throughthe barrier only “as a wave”.

The TE has a wide range of applications, however it is mainlyused in the electronic field. We owe to Esaki (1957) the constructionof the first “tunnel diode”, which he made working on germaniumsemiconductor diodes, characterized by a very thin thickness of thejunction. Esaki realized that electrons were able to penetrate verythin barriers (with a thickness around one hundred atoms) digginga tunnel in the layer they had to go through. It was a completelynew phenomenon, impossible according to the laws of the classicalphysics. Esaki’s diode has been widely used with amplifiers, oscillators,trigger circuits, electronics switches. It was Josephson (1962) [10] whocreated a particular junction in semiconductor materials (JosephsonJunction), which allows the transit of the electrons through the TE,with low consumptions and without releasing too much heat. “A veryinteresting situation was noticed by Josephson while analyzing whatmight happen at a junction between two superconductors. Supposewe have two superconductors which are connected by a thin layer ofinsulating material. Such an arrangement is now called a Josephsonjunction. If the insulating layer is thick, the electrons can’t getthrough; but if the layer is thin enough, there can be an appreciablequantum mechanics amplitude for electrons to jump across. Thisis just another example of the quantum-mechanical penetration of

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a barrier. Josephson analyzed this situation and discovered that anumber of strange phenomenon should occur” [7]. Josephson junctionis very much used in semiconductors, i.e. in the memory cells of thecomputers. The last use of the TE is at the IBM of Zurich, a TEelectronic microscope.

Though calculations confirm that such a peculiar phenomenon, asthe TE, can occur, we do not know exactly how it actually happens;that is in which way it takes place. It is widely thought that, througha quantum fluctuation mechanism, the particle borrows the lackingenergy, respect to the potential of energy to get through. As soonas the particle goes through the barrier, it will immediately releasethe mentioned energy. This mechanism is allowed by the quantummechanics, the shortest the span of the energetic loan, the higher theprobabilities that this phenomenon can occur. All clear! What is notclear at all is how the particle materially gets through the barrier.The borrowed energy can be worth both for the electrons and for thehadrons. The TE is at the bottom of some radioactive phenomena,such as α emission. That is even two protons and two neutrons,bound in one helium nucleon, thanks to the TE manage to get freefrom the nucleon with a high atomic weight which they belong to.This phenomenon, however, occurs very rarely, as we learn from theradioactive half time tables. Indeed, “since the particle α has twoprotons, it is pushed back by the whole charge of the protons thus theparticle tries to escape but the wall around the nucleon prevents theparticle from getting free. In this way the particle is trying to makea tunnel. It wants to penetrate the barrier because it wants to escapeand of course, sooner or later it will manage to do that. How longwill that take? Maybe some thousand of years, but we cannot be sureabout that. It may probably take it thousands of years to escape, butit may also get free in any moment. There is no way to be sure aboutit: it is just a matter of probability” [3].

However, there must be a reason if the TE is more likely to happenwith lighter particles such as the electron rather than with bigger ones.Someone may say that this happens because, compared to heavierparticles, “the very small size of the electron implies that it undergoes agreat influence of its associated wave. It is the undulation nature of theelectron to allow it to go through the barrier” [11]. Apparently it is thevery small mass of the electron to allow it to have a wider associatedwave, this makes its undulation behaviour, as well as the TE, easier.On the contrary “those particles having a much bigger mass than theelectron, have a much shorter associated wave” [12]. This limits alot its potential undulation behaviour, which can be verified by a TErarer than the electron and more and more infrequent as the mass of the

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particle increases. Yes, the electron may have an undulation behaviourbut it is still a “matter wave”; it has a mass of 0,511/c2 MeV, that is9, 11 · 10−28 gr It will be said: the lacking energy is borrowed througha quantum fluctuation mechanism. That’s all right! Thus the electronwill go through the barrier as a wave. It is clear but only till a certainextend. What happens to the mass while the particle goes through thelayer? It could be said: the particle takes it with itself; besides theelectron is the lightest particle having a mass. Right, but it still has amass, it occupies a certain surface: how can it go through a barrier?But if we consider the electron as having 2 (or 3) smaller particles,it will be easier to understand this phenomenon. Maybe a particlewith a mass half (or one third) of the electron may be able to passthrough the barrier more easily. The mechanism should be analogousto what may happen with the I. That is, with the TE the electronmay penetrate the barrier as a “field” (the electron field), separatingits LQs (or rather getting “spread” with its LQs), in order to reducethe impact surface which will collide against the layer, in this wayincreasing the probabilities to find a passage through the “meshes” ofthe wall.

It is useful at this point to make a consideration. When we earliertried to measure the mass of the probable LQs, we divided the massof the electron and what may be the number of particles which couldmake the electron. However it may be not accurate and safe to followthis method, it might take us on the wrong way. If we measure themass of what makes the proton (2 up Qs and 1 down Q), we get thevalue of 12 MeV. Yet, the mass of the proton is 938,9 MeV! That is thewhole mass of the 3 Qs making the proton is only about one eightiethof the mass of the proton. In the same way (very indicatively), if theelectron is made of LQs, we may have that the total mass of the LQsof every single electron may represent only about one eightieth of themass of the electron. But we have to suppose that there may be 2 (oreven three) LQs for each electron. Thus we should divide the massof the electron for 160 (or 240). In the first case (maybe the mostprobable) we have: 511000 eV/160 = 3193,75 eV. That is, if there are2 LQs, for each of them we should have a mass of 3,193 KeV. In thesecond case we would have: 511000 eV/240 = 2129,166 eV. That is, ifthere are three LQs for each electron, we have a mass of 2,129 KeV (itis implicit that every time we express the mass in energetic equivalent,at the denominator it is understood c2, that is the square of the speedof light).

Incidentally, we need to consider that where the electron weights0,511 MeV, the Q up (the lightest) weights 3 MeV (a MeV is a million ofelectron volt), and the Q down is 6 MeV. This means that an electron

Progress In Electromagnetics Research, PIER 58, 2006 219

weights 1/6 of a Q up and 1/12 of a Q down. Hence, in case there areonly two LQs which make the electron, we have that one LQ weightsone thousand of the lightest Q (and about 1/320000 of the mass of theproton).

Why all this? What is the use of these calculations and rates?Because a very small and light LQ, probably just 1/160 of the mass ofthe electron, or even something like one thousand of the weight of a Qup, makes it “closer” to a wave, it increases the probabilities to “finda path in the layer” to pass through. This makes more acceptableand/or likely what we thought. If we think about it, the energy ofa probable LQ, which may approximately correspond to 3200 eV, isjust the energy of a “soft” x ray. That is a LQ would have the same“size”, at least energetic, of “less penetrating” x rays. Moreover, tothat energy will have always to correspond a determined and specificfrequency and wave length. As Giacconi and Tucker remind us, “x rayshave the right wavelength to see the microscopic world” [13]. Thus,we may say, the electronic microscope allows us a vision as with xrays. It seems really an analogy, a real correspondence, between thebehaviour of the electron in its all, and an x ray. That is a LQ isapparently superimposing with an EMW of a certain frequency. Thus,the energy of a LQ could explain its undulation behaviour, how theelectron manages to appear and behave like a wave too.

Besides, the dimension of the particle must have a great meaningto the fulfilment of the TE. To confirm this we have the great differencein probability that this phenomenon may happen: it is more likely forthe electron (0,511 MeV), and less likely for the proton or neutron(respectively 938 and 940/c2 MeV). It would take some thousand yearsto a particle α to pass spontaneously through a barrier: Its mass is6, 6 · 10−24 gr. If we convert it in MeV, we have that the mass lost toget the fusion of a helium nucleon is about 0,63% of the sum of theparticle masses making it (two nuclei of deuterium) [12], that is themass of a helium nucleon is about 3735 MeV. It could be objected: let’sconsider that the electron may go through the barrier as we assumed,but how can a baryon, or even a particle α, to penetrate the barrierin the same way of an electron? Well, the hadrons are made of Qs,this is not a hypothesis, it is verified! Thus it is possible that Qs usethe same mechanism foreseen for the LQs. In this case the biggerdifference in weight (Qs are certainly heavier than the hypotheticalLQs), may explain why a particle made of Qs passes through a barrierwith less probabilities than a LQ. Besides, we should also consider thedifference in strength between the colour force and the LF. It should bemuch more likely that Qs (within their limits of movement) are kept“in check”, thus it is very difficult that a hadron has an undulation

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behaviour, indeed, as we said earlier, it has a “much shorter associatedwave” [12].

3. CONCLUSIONS

Summing up, there is a still mysterious phenomenon: the I induced byelectron. It is not explainable from the point of view of the classicalmechanics, but it is solved from a mathematical point of view, using theformalism of the quantum mechanics and applying to the phenomenonthe solution of Schrodinger’s equation. The quantum solution to theproblem is clear and elegant, especially from a mathematic point ofview, but it leaves us some perplexity as for the understanding of thereal fulfilment of the phenomenon.

It has been expressed a hypothesis to understand the I inducedby the electron: it is really a peculiar phenomenon. According toQuantum Mechanics “when we talk about a situation as the electrongoing through the slits, we describe it with an amplitude. It issomething similar to the waves; it is often called wave function. Theamplitude can go through both slits and produce I, just as the waterwaves. But then where are the particles? Which slit do they actuallygo through? The amplitude does not tell you anything about it” [3].Maybe if we consider that the electron might be made of smallerparticles, just as the baryons and mesons, (safeguarding the integrity ofthe unitary negative charge, the conservation of the leptonic number,the conservation of the angular momentum etc.), this may help usunderstand how a single electron can go through two close holes atthe same time. Besides, there is another very particular quantumphenomenon, which is explained with the quantum mechanics too,and which gives us some perplexities about how it really happens,that is the TE. The energy borrowed through spontaneous quantumfluctuations, though justifying the energetic compensation given infavour of the electron (or another particle) and allowing it to go throughthe barrier more easily, does not clarify how actually the electronpierces the barrier.

It can be easier to understand the TE phenomenon if we considerthat it may have the same mechanism of the I. That is, if it is possibleto divide the electron in smaller particles, as to be superimposing witha low energy x ray (soft x ray), as a consequence the reduction ofthe surface of the particle colliding with the barrier could make thepenetration of the barrier itself easier.

On the other hand, it should not appear too unreal the idea thata LQ can be superimposing with an x ray, both as analogous energeticvalue, and in its real behaviour, that is as a wave. We all know

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that the electron too is superimposing with an electromagnetic wave,though more energetic than an x ray: a γ ray in its case. Indeed,just as suggested by Dirac, in the collision between an electron andits antiparticle, there is the annihilation of the particles themselves,with a complete transformation of their mass in a couple of γ rays,having an energy exactly equal to the combined mass of the annihilatedparticles” [12]. Therefore, to an electron having a mass of 9, 11·10−28 grwhich equivalent energetic is 0,511 MeV, will have to correspond anelectromagnetic wave of 5, 11·105 eV. Now, since in some circumstanceswe can consider the electron equivalent to a γ ray of a definite energy,we should be able to explain its undulation behaviour: both when theelectron induces I, typical of the waves, and when the electron utilizesthe TE (a phenomenon much easier to make for a wave than for aparticle). Thus, there should not be any more any reason to introducethe concept of the LQs. The observation is right, though it does not tellus yet how a single particle manages to pass simultaneously throughtwo holes: it seems more logic and convincing that in this case theelectron splits, for a moment, in its probable components.

This is a willing attempt to understand some of the mostunexplainable quantum phenomena. It is not possible to demonstratethis hypothesis; we should wait for more powerful colliders, such as theLarge Hadron Collider, which is supposed to be prepared in Genevein three years. Beyond any disprove or confirmation, the concept weshowed should not be in contrast with what we know from Physics.

ACKNOWLEDGMENT

We warmly thank Dr. Salvatore Verdoliva for the translation of thispaper and for his precious collaboration and support.

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